Timeline for Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?
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14 events
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| Sep 22, 2023 at 0:19 | comment | added | Zhi-Wei Sun | Yaakov, thank you for the clarification. So, Question 2 remains open as 8558169401 is not a counterexample to my conjecture. | |
| Sep 20, 2023 at 16:45 | comment | added | Yaakov Baruch | I get $8558169401=x(3x+1)/2+y(7y+1)/2+2^k$ for $x=51199, y=-36356, k=1$ or $x=34425, y=36383, k=31$, among more than 20 other solutions. | |
| Apr 18, 2021 at 14:17 | comment | added | Will Sawin | Yes. I currently think this conjecture is wrong, but the first counterexamples just might be very large. | |
| Apr 18, 2021 at 14:12 | comment | added | Zhi-Wei Sun | @Sawin Please note my conjecture on writing positive integers as $2^x+\lfloor y^2/3\rfloor+\lfloor z^2/4\rfloor$. Clearly, $$\{\lfloor y^2/3\rfloor:\ y\in\mathbb N\}=\{3y^2:\ y\in\mathbb N\}\cup\{y(3y+2):\ y\in\mathbb Z\}$$ and $$\{\lfloor z^2/4\rfloor:\ z\in\mathbb N\}=\{z^2:\ z\in\mathbb N\}\cup\{z(z+1):\ z\in\mathbb N\}.$$ | |
| Apr 18, 2021 at 13:58 | comment | added | Will Sawin | I would guess for probabilistic reasons that no finite set of expressions of the form quadratic polynomial + an exponential covers all positive integers. However, small-number effects might imply that the first counterexample is quite large. If you want to find evidence against the probabilistic model / evidence for your conjecture, it might be worthwhile to calculate the mean and variance of the number of $k$ such that there exist $x$, $y$ representing $k$. If the variance is much less than the mean then the probabilistic model is likely oversimplified. | |
| Apr 18, 2021 at 12:22 | comment | added | Zhi-Wei Sun | Giovanni Resta has just informed me that he has found that 8558169401 is the first positive integer not of the form $x(3x+1)/2+y(7y+1)/2+2^k$. On the other hand, concerning my conjecture that any positve integer can be written as $x^6+y^3+z(3z+1)/2+2^k$ $(k,x,y\in\mathbb N,\ z\in\mathbb Z)$, he has verified it for $n$ up to $10^{10}$ without counterexamples found, see oeis.org/A343460 . Note that $1/2+1/3+1/6=1$. | |
| Apr 18, 2021 at 2:52 | comment | added | Zhi-Wei Sun | In contrast with the questions, I conjectured that any integer $n>1$ can be written as $2^x+\lfloor y^2/3\rfloor+\lfloor z^2/4\rfloor$ with $x,y,z\in\mathbb Z$, and this was verified for $n$ up to $10^{10}$; see oeis.org/A343397 . | |
| Apr 18, 2021 at 2:48 | comment | added | Zhi-Wei Sun | The oeis sequence has been created, but not yet approved by the editors. | |
| Apr 18, 2021 at 2:47 | comment | added | Will Sawin | The oeis link does not appear to be working yet. | |
| Apr 18, 2021 at 2:36 | comment | added | Will Sawin | The probability a number $n$ is represented by a quadratic polynomial should be proportional to $1/\sqrt{\log n}$, so the number of $k$ so that $n -2^k$ is represented should be proportional to $\sqrt{\log n}$, which if the statistics are Poisson-like would mean the probability $n$ cannot be represented is $e^{ - O( \sqrt{ \log n})}$, whose sum is infinte, suggesting that many numbers can't be represented. But maybe statistics are not Poisson-like... | |
| Apr 18, 2021 at 2:03 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Apr 18, 2021 at 1:55 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Apr 18, 2021 at 1:49 | history | edited | Zhi-Wei Sun | CC BY-SA 4.0 |
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| Apr 18, 2021 at 1:42 | history | asked | Zhi-Wei Sun | CC BY-SA 4.0 |